typically associated with conservation of “mass”, “momentum” and “energy”.
Let \(u(x,t)\)  describe the random KdV wave field in a soliton gas – the “integrable turbulence” [1]}. In the spirit of the Whitham modulation theory [2]} we invoke scale separation and, assuming ergodicity of the soliton gas, apply the ensemble averaging to the KdV conservation laws (REF ) to obtain the modulation system (note however that the original Whitham theory uses spatial averaging)
\(\frac{\partial }{\partial t} \langle P_i[u ] \rangle + \frac{\partial }{\partial x} \langle Q_i[u] \rangle = 0, \quad i=0, 1, 2, \dots \) 
